Electricity and Magnetism
• Review
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Electric Charge and Coulomb’s Force
Electric Field and Field Lines
Superposition principle
E.S. Induction
Electric Dipole
Electric Flux and Gauss’ Law
Electric Potential Energy and Electric Potential
Conductors, Isolators and Semi-Conductors
Feb 27 2002
Today
• Fast summary of all material so far
– show logical sequence
– help discover topics to refresh for Friday
Feb 27 2002
Electric Charge and
Electrostatic Force
• New Property of Matter: Electric Charge
– comes in two kinds: ‘+’ and ‘-’
• Connected to Electrostatic Force
– attractive (for ‘+-’) or repulsive (‘- -’, `++’)
• Charge is conserved
• Charge is quantized
Feb 27 2002
Elementary Particles
10-20
Strength
10-7
Weak Force
Atomic Nuclei
10-15
Strong Force
Atoms
Molecules
10-10
10-5
Human
Electric Force
100
1
1
5
Earth 10
Solar System
1010
1015
Feb 27 2002
Farthest
Galaxy
1020
Gravity
10-36
Coulomb’s Law
• Inverse square law (F ~ 1/r2)
• Gives magnitude and direction of Force
• Attractive or repulsive depending on
sign of Q1Q2
Feb 27 2002
Coulomb’s Law
F12
Q1
r21
r21
Feb 27 2002
Q2
Coulomb’s Law
Q1
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r21
F12
r21
F12 = - F 21
Q2
Superposition principle
Q3
F13
Q1
F12
F1,total
Q2
• Note:
– Total force is given by vector sum
– Watch out for the charge signs
– Use symmetry when possible
Feb 27 2002
Superposition principle
• If we have many, many charges
– Approximate with continous distribution
• Replace sum with integral!
Feb 27 2002
Electric Field
• New concept – Electric Field E
• Charge Q gives rise to a Vector Field
• E is defined by strength and direction of
force on small test charge q
Feb 27 2002
The Electric Field
• Electric Field also exists is test charge q is not
present
• The charge Q gives rise to a property of space
itself – the Electric Field
• For more than one charge -> Superposition
principle
Feb 27 2002
Electric Field
• For a single charge
+Q
• Visualize using Field Lines
Feb 27 2002
Field Lines
• Rules for field lines
– Direction: In direction of E at each point
– Density: Shows magnitude of E
– Field Lines never cross
– From positive to negative charge
• i.e. show direction of force on a positive charge
– Far away: Everything looks like point charge
Feb 27 2002
Electric Dipole
Torque τ = p x E
p = Q l Dipole moment
Feb 27 2002
Electrostatic Induction
+ ++ -- ++
+ +
-- +
+ +
+
+
+ +
-+
++
Feb 27 2002
• Approach neutral object with
charged object
• Induce charges (dipole)
• Force between charged and
globally neutral object
Electric Flux
•
•
•
•
Electric Flux: ΦE = E A
Same mathematical form as water flow
No ‘substance’ flowing
Flux tells us how much field ‘passes’ through
surface A
Feb 27 2002
Electric Flux
• For ‘complicated’ surfaces and non-constant E:
– Use integral
• Often, ‘closed’ surfaces
Feb 27 2002
Electric Flux
• Example of closed surface: Box (no charge inside)
dA
dA
E
• Flux in (left) = -Flux out (right):
Feb 27 2002
ΦE = 0
Gauss’ Law
• How are flux and charge connected?
• Charge Qencl as source of flux through closed
surface
Feb 27 2002
Gauss’ Law
• True for ANY closed surface around Qencl
• Relates charges (cause) and field (effect)
Feb 27 2002
Gauss’ Law
• Different uses for Gauss’ Law
– Field E -> Qencl (e.g. conductor)
– Qencl -> Field E (e.g. charged sphere)
• Proper choice of surface – use symmetries
Feb 27 2002
Hollow conducting Sphere
++ + +
+
+
+
+
+
+
+
++
+
+ + +
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++ + +
+
+
+
+
+
+
+
++
+
+ + +
Gauss’ Law
• Charge Sphere radius r0, charge Q, r > r0
r0
Q
r
Qencl = Q
Feb 27 2002
dA
Gauss’ Law
• Most uses of Gauss’ Law rely on simple
symmetries
– Spherical symmetry
– Cylinder symmetry
– (infinite) plane
• and remember, E = 0 in conductors
Feb 27 2002
Work and Potential Energy
F(l)
dl
b
x
α
x
a
Work:
Conservative Force:
Potential Energy
Feb 27 2002
Electric Potential Energy
• Electric Force is conservative
– all radial forces are conservative (e.g. Gravity)
• We can define Electric Potential Energy
F
Feb 27 2002
Example: Two charges
Q
q
r
• If q,Q same sign:
– U > 0; we have to do work ‘pushing’
charges together
• If q,Q unlike sign:
– U < 0; Electric force does work ‘pulling’
charges together
Feb 27 2002
Electric Potential
• Electric Potential Energy proportional to q
• Define V = U/q
• Electric Potential V:
– Units are Volt [V] = [J/C]
Feb 27 2002
Electric Potential
• Note: because V = U/q -> U = V q
– for a given V: U can be positive or negative,
depending on sign of q
• V :Work per unit charge to bring q from a to b
• Ex.: Single Charge
Feb 27 2002
Electric Potential for many
charges
• Superposition principle....
V(x) = Σ1/(4πε0) Qi/ri
• Sum of scalars, not vectors!
• Integral for continous distributions
Feb 27 2002
Example: Three charges
Q2
Q3
Q1
r2
r3
r1
V(x) = Σ1/(4πε0) Qi/ri
x
Feb 27 2002
Example: Capacitor plates
+
+
+
+
+
+
+
+
+
+
a
+q
b
xa
xb
x=0
Feb 27 2002
x=d
x
Example: Capacitor plates
+
+
+
+
+
+
+
+
+
+
a
+q
xa
x=0
Feb 27 2002
b
xb
x=d
V(x)
0
U(x,q)
x
q<0
x
x
q>0
Applications
+
+
+
+ Velocity v
++q
+
+
+
+
+
Feb 27 2002
d
-
• Energy for single particle
(e.g. electron) small
• Often measured in
‘Electron Volt’ [eV]
• Energy aquired by
particle of charge 10-19 C
going through ∆V=1V
Conductors
• E = 0 inside
– otherwise charges would move
• No charges inside
– Gauss
• E perpendicular to surface
– otherwise charges on surface would move
• Potential is constant on conductor
Feb 27 2002